Limits and Continuity: Foundations for Calculus

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Secant and Tangent Lines; The Idea of Limits

Secant and Tangent Lines

Secant and tangent lines are fundamental concepts for understanding the behavior of functions and the concept of limits.

Secant Line: A line passing through two distinct points on the graph of a function f.
Tangent Line: A line that touches the graph of f at exactly one point (x = a), representing the instantaneous rate of change at that point.
Slope of Secant Line:
Slope of Tangent Line:

Example (Velocity): If s(t) is the position of an object at time t:

Average Velocity:
Instantaneous Velocity:

Graph of position function s(t) with secant lines illustrating average velocity

Definitions and Properties of Limits

Limit of a Function

The limit of a function describes the value that f(x) approaches as x approaches a specific value a. It is written as .

The limit depends on the values of f(x) near x = a, not necessarily at x = a.
Limits can be evaluated graphically, numerically (using tables), or algebraically.

One-Sided Limits

Right-sided limit: (as x approaches a from the right)
Left-sided limit: (as x approaches a from the left)
If the left- and right-sided limits are not equal, the two-sided limit does not exist (DNE).

Techniques for Computing Limits

Limit Laws

Limits can be evaluated using algebraic properties:

Sum/Difference:
Product:
Quotient: (if denominator ≠ 0)
Power/Root:

Direct Substitution

For polynomials and rational functions (where denominator ≠ 0), substitute x = a directly.
If substitution yields (indeterminate form), use factoring, conjugates, or other algebraic techniques to simplify.

Squeeze Theorem

If near , and , then .

Graph illustrating the Squeeze Theorem with bounding functions

Oscillating Functions and the Squeeze Theorem

Some functions oscillate as x approaches a value. The Squeeze Theorem can be used to show that the limit exists even if the function itself oscillates.

Graph of y = x^2 sin(1/x) squeezed between y = x^2 and y = -x^2

Infinite Limits and Vertical Asymptotes

Infinite Limits

An infinite limit occurs when f(x) increases or decreases without bound as x approaches a value a:

means f(x) grows arbitrarily large.
means f(x) decreases without bound.

Graph showing f(x) growing arbitrarily large as x approaches a Graph showing f(x) becoming arbitrarily large negative as x approaches a

Vertical Asymptotes

A vertical asymptote occurs at if approaches or as approaches from at least one side.

Graphical Identification of Infinite Limits

To evaluate infinite limits from a graph, observe the behavior of as approaches the value of interest from the left and right.

Graph of a function with vertical asymptotes at x = 2 and x = 4

Limits at Infinity and Horizontal Asymptotes

Limits at Infinity

Limits at infinity describe the end behavior of a function as becomes very large (positive or negative):

means approaches as increases without bound.
means approaches as decreases without bound.
Horizontal asymptotes occur at and .

Graph showing horizontal asymptotes as limits at infinity

Continuity

Continuity at a Point

A function f is continuous at if:

is defined
exists

Types of Discontinuities

Removable discontinuity: A hole in the graph; the limit exists but is undefined or not equal to the limit.
Jump discontinuity: The left- and right-hand limits exist but are not equal.
Infinite discontinuity: The function increases or decreases without bound near the point (vertical asymptote).
Oscillating discontinuity: The function oscillates near the point without approaching a single value.

Graph of a function with various types of discontinuities

Continuity on Intervals

A function is continuous on an interval if it is continuous at every point in the interval. At endpoints, one-sided continuity is used.

Summary Table: Types of Asymptotes and Limits

Type	Definition	Equation	Graphical Feature
Vertical Asymptote	Function grows without bound as x approaches a	x = a	Infinite limit
Horizontal Asymptote	Function approaches a finite value as x → ±∞	y = L	Limit at infinity
Slant Asymptote	End behavior approaches a non-horizontal line	y = mx + b	Degree numerator = degree denominator + 1

Additional info: This guide covers foundational concepts in limits and continuity, including graphical and algebraic techniques, and the classification of discontinuities and asymptotes. These topics are essential for further study in calculus and mathematical analysis.